Digital wireless communication can be described as a class of methods used to transmit data (or message) in the form of numbers and/or symbols from a source point and receive the data at a remote destination point. Many satellite, airborne, and terrestrial wireless systems use a digital form of communication. The digital form of communication can require that the data be transformed into symbols. The set of symbols typically used is usually small, e.g., consisting of two, four, or eight different symbols.
A larger volume of data can be represented as an ordered group (i.e., a sequence) of symbols. Each data symbol can be represented as one or more physical quantities at different points in a circuit chain of a transmitter, through a medium of air and/or free space, and in a circuit chain of a receiver, until the data is converted back to a data symbol. Some of the physical quantities can be functions over a time period.
In particular, through the medium, a data symbol can take the form of an electromagnetic wave lasting over a symbol with a particular frequency, strength, and a starting phase angle. The medium and the electromagnetic spectral band used can be together known as the communication channel. Throughout the communication system, many of the intermediate quantities corresponding to a symbol can be called signals to distinguish them from the data symbol.
These signals can experience deterministic and nondeterministic transformations as they advance from one point to the next. When the signal arrives at the receiver circuit chain, it is usually corrupted by noise and/or other forms of interference. Such interferences are examples of nondeterministic transformations that communication signals experience. The extent of interference can also be known as the level of degradation of the channel. Interference can be intentional or unintentional. Jamming a communication system by directing electromagnetic energy at the receiving antenna is an example of interference intended to disrupt communication. A channel affected by intentional interference is also known as a contested channel.
Wireless communication typically uses forward error correction (FEC) encoding to combat interference and improve the reliability (or probability) of recovering (or reconstructing) the transmitted message. FEC encoding can introduce redundant symbols (that are functions of the original data symbols) at the transmitter and typically uses a corresponding decoding scheme at the receiver. The ratio of the number of original data symbols to the total number of original plus redundant symbols is known as the coding rate. Half and one third rate FEC are common. An additional technique to improve the recovery of the original message to be transmitted over a degraded channel is to transmit multiple copies of symbols (e.g., copies of FEC encoded symbols) interspersed over a time period. A simple reasoning of how this can improve the probability of recovering the symbol is that at least one of the copies may escape interference or may be only mildly affected by interference. Such a communication method is said to use “repeat codes.” Repeating FEC encoded symbols can reduce the overall coding rate. At the receiver, signals corresponding to multiple copies of a symbol should be properly combined. In an example, let the FEC encoded data symbols needed to be transmitted be:                1, 4, 2, 2, 3, 1        
At some points in the communication system, signals corresponding to a symbol can be complex numbers.
In this example, let the signals corresponding to the above six symbols be the following complex numbers:                1, −i, i, i, −1, 1at the transmitter. In the above, i denotes the unit imaginary part √{square root over (−1)} of complex numbers. This transformation from the symbols to the signals is an example of the quadrature phase shift keying (QPSK) modulation scheme.        
Let the transmitter transmit the above symbol sequence two different times in the “repeat codes” mode. The signals can experience deterministic and nondeterministic transformations as they advance through the communication system. Let the original sequence of signals be multiplied by the complex number 0.98+0.19i. The resulting signals after this transformation are:                0.98+0.19i, 0.19−0.98i, −0.19+0.98i, −0.19+0.98i, −0.98−0.19i, 0.98+0.19iat some point in the receiver. Let the copy of the signals experience a slightly different transformation and be multiplied by the complex number 0.92+0.39i instead of being multiplied by the earlier factor 0.98+0.19i. The resulting signal sequence of the repeated transmission after this transformation is:        0.92+0.39i, 0.39−0.92i, −0.39+0.92i, −0.39+0.92i, −0.92−0.38i, 0.92+0.39i        
In addition to such transformations, let the signals at the receiver be corrupted by noise over and above the transformations. Let the resulting transformed and noisy signals for the original and the repeat transmissions be                0.80+0.16i, 0.24−1.04i, −0.38+0.92i, −0.34+1.11i, −0.79−0.24i, 1.10+0.33iand        0.87+0.31i, 0.53−0.89i, −0.43+1.05i, −0.24+0.86i, −0.83−0.45i, 0.77+0.32irespectively. In this example, combining the two received signals sequences can be accomplished by averaging the two complex signal sequences. The resulting combined signal sequence is:        0.84+0.24i, 0.38−0.97i, −0.41+0.99i, −0.29+0.99i−0.81−0.35i, 0.94+0.33irounded off to two decimal places.        
In the above example, the multiplication of the signal sequence corresponding to the original transmission by the complex number 0.98+0.19i and the multiplication of the signal corresponding to the repeat transmission by the complex number 0.92+0.39i can be due to the phase shifts that the signal sequences experience.
The phase shifts are examples of the above mentioned deterministic transformations in the sense that all the signals in a sequence are multiplied by the same complex number and the receiver can “anticipate” that such a multiplication takes place. It is a nondeterministic transformation in the sense that the receiver may not know the exact complex number that the signal sequence gets multiplied by. In some communication system, the circuits in the receiver can track these phase shifts fairly accurately and make the needed correction. Examples of conditions that can facilitate accurate phase tracking are (1) constant carrier frequency, (2) strong direct communication path and negligible multipath reflections of the electromagnetic waves reaching the receiver, (3) good atmospheric conditions contributing to the stability of the carrier frequency phase angle, and/or (4) adequate signal to noise ratio (SNR). Under these conditions, a system can transmit reference symbols of known values only occasionally to help the receiver with accurate phase tracking. The receiver knows the locations and values of reference symbols and uses the corresponding signals for phase tracking.
Examples of systems that do not operate with such favorable features are (1) frequency hopped systems that change the carrier frequency frequently to avoid being intercepted and also to reduce the level of channel degradation caused by narrow band jammers, and/or (2) systems operating with possible multipath receptions, with multiple antennas, and over a degraded channel. Many such systems are designed to transmit reference symbols frequently. One problem is that the reference symbols are also corrupted by noise and an estimate of the phase shift computed with the use of reference signals only can be inaccurate. Data signals carry some information about the phase shift and the receiver can use them in conjunction with the reference signals to improve the accuracy of phase shift estimation.
The problem of estimation of phase shift is compounded if the receiver can combine multiple copies of an original transmission. Systems that incorporate systematic repeat codes are not the only ones in which the receiver can take advantage of combining. There are communication systems that operate with a data link protocol in which a receiver requests a repeat if an originally received signal sequence could not be successfully converted to data symbols. Ordinarily, the receiver discards the original reception and processes only the repeated version. A proper combining of the multiple copies can be a better option. Systems that use multiple antennas can also take advantage of combining the multiple copies received from multiple antennas arising out of a single transmission.
The process of determination of symbol decisions from the received signals can be called data symbol detection (or just “detection”). Phase shift estimation and detection can be treated as a joint problem to improve the probability of correct detection and it is desirable to do so.